494 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xvm 



Then 



4abcd x 2cd a 2 +b 2 y c 2 +d 2 



~~ "~" " 



' z z 2 2ab z 2cd 



The problem to choose a, , d so that a 2 +& 2 = c 2 +d 2 and so that the 

 expression for z 2 shall be a square was treated by Euler in 3-17. In 

 18, he began by setting (in accord with the above) 



, 



2ab ' 2cd 



Then Al = (a6) 2 /(2a6), BI = (cd=d) 2 /(2cd). Hence the conditions are 



4abcd 

 Make the numerators the squares of r 2 +s 2 and p 2 -\-q 2 by setting 



To make the common denominator a square, we have the condition 



which is satisfied if we have two rational right triangles the ratio of whose 

 areas is a square [cf. Euler 81 of Ch. XVI]. The above ratio is a/0 for 

 p = 3a, g = 20 a, r = 30, s = 2a and for seven similar sets. The case 

 a = 9, = 4 gives p = 27, q = 1, r = 12, s = 14. By a table (p. 60) of values 

 of xy(x 2 y 2 ), we get right triangles of equal areas 2-3-5-7 for x = 5, y = 2; 

 x = 6,y = l', x = 8,y = 7', also two of equal area for 



r = p = m 2 -i-inn-\-ri 2 ) q = m 2 n z , s = n z -}-2'mn. 



Euler 19 made the four expressions ABAB all squares. Set A = xfz, 

 B = y/z. Then xy.z(x-\-y} and xyz(xy) shall be squares. This will 

 be the case if 



xy = a 2 +b =c 2 +d z , z(x-\-y) =2ab, z(xy}=2cd } 

 whence 



ab-\-cd ab cd a?b 2 c 2 d 2 



T = II = ? 2 = 



z z a 2 +b 2 



Since xy shall be a sum of two squares in two ways, set 



a = pr-\-qs, b = ps qr, c = pr qs, d = ps-\-qr. 

 Then 



2rs(p 2 q~) 2pq(r~ s 2 ) 4pqrs(p 2 q~}(r 2 s 2 ) 



19 Novi Cornm. Acad. Petrop., 15, 1770, 29; M6rn., 11, 1830 (1780), 31; Comm. Arith., I, 

 414; II, 438. The simpler solution here reproduced is given in the second of these two 

 papers, and is practically the same as that in Euler's posthumous paper, Comm. Arith., 

 II, 586-7; Opera poaturna, 1, 1862, 137-9. In two letters to Lagrange (Oeuvres, XIV, 

 214, 219), Jan. and March, 1770, Euler (Opera postuma, 1, 1862, 573-4) gave discussions 

 occurring in the first and third of these papers. On the related problem to find p, q, r, s 

 such that \pqrs(p 4 s i )(q*r t ) D, see Euler, Opera postuma, I, 487^490 (about 1766). 

 The second letter is quoted in I'interm6diaire des math., 21, 1914, 129-131, and in 

 Sphinx-Oedipe, 7, 1912, 57-8. First paper in Opera Omnia, (1), III, 148. 



